From 32a3ee96ae4ae5a54b82365491a40edfc2212494 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sat, 23 Oct 2021 22:46:11 +0200 Subject: Writing --- ising_scaling.tex | 28 +++++++++++++++++----------- 1 file changed, 17 insertions(+), 11 deletions(-) (limited to 'ising_scaling.tex') diff --git a/ising_scaling.tex b/ising_scaling.tex index c7e7adb..04cc0eb 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -137,16 +137,21 @@ to constant rescaling of $u_h$). The invariant scaling combinations that appear as the arguments to the universal scaling functions will come up often, and we will use $\xi=u_h|u_t|^{-\Delta}$ and $\eta=u_t|u_h|^{-1/\Delta}$. -The analyticity of the free energy at places away from the critical point implies that the functions -$\mathcal F_\pm$ and $\mathcal F_0$ have power-law expansions of their -arguments about zero. For instance, when $u_t$ goes to zero for nonzero $u_h$ -there is no phase transition, and the free energy must be an analytic function -of its arguments. It follows that $\mathcal F_0$ is analytic about zero. This -is not the case at infinity: since $\mathcal F_0(\eta)=\eta^{D\nu}\mathcal -F_\pm(\eta^{-1/\Delta})$ has a power-law expansion about zero, $\mathcal -F_\pm(\xi)\sim \xi^{D\nu/\Delta}$ for large $\xi$. The nonanalyticity of -these functions at infinite argument can therefore be understood as an artifact -of the chosen coordinates. +The analyticity of the free energy at places away from the critical point +implies that the functions $\mathcal F_\pm$ and $\mathcal F_0$ have power-law +expansions of their arguments about zero, the result of so-called Griffiths +analyticity. For instance, when $u_t$ goes to zero for nonzero $u_h$ there is +no phase transition, and the free energy must be an analytic function of its +arguments. It follows that $\mathcal F_0$ is analytic about zero. This is not +the case at infinity: since +\begin{equation} + \mathcal F_\pm(\xi) + =\xi^{D\nu/\Delta}\mathcal F_0(\pm \xi^{-1/\Delta})+\frac1{8\pi}\log\xi^{2/\Delta} +\end{equation} +and $\mathcal F_0$ has a power-law expansion about zero, $\mathcal +F_\pm$ has a series like $\xi^{D\nu/\Delta-j/\Delta}$ for $j\in\mathbb N$ at +large $\xi$, along with logarithms. The nonanalyticity of these functions at +infinite argument can be understood as an artifact of the chosen coordinates. For the scale of $u_t$ and $u_h$, we adopt the same convention as used by \cite{Fonseca_2003_Ising}. The dependence of the nonlinear scaling variables on @@ -210,7 +215,8 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant. To lowest order, this singularity is a function of the scaling invariant $\xi$ alone. It is therefore suggestive that this should be considered a part of the singular free energy and moreover part of the scaling function that composes -it. We will therefore make the ansatz that +it. There is substantial numeric evidence for this as well. We will therefore +make the ansatz that \begin{equation} \label{eq:essential.singularity} \operatorname{Im}\mathcal F_-(\xi+i0)=A_0\Theta(-\xi)\xi e^{-1/b|\xi|}\left[1+O(\xi)\right] \end{equation} -- cgit v1.2.3-70-g09d2